Final answer:
The transformation of f(x) to h(x) consists of shifting the graph to the left by 5 units and downward by 2 units. The function h(x) is found to be 3(x + 5)² - 4, and evaluating h(-3) yields a value of 8.
Step-by-step explanation:
Part A: Transformation Description
The transformation from f(x) to h(x) = f(x + 5) - 2 can be described as a horizontal shift to the left by 5 units and a vertical shift downward by 2 units. This is due to the fact that adding a positive number inside the function argument (x + 5) shifts the graph to the left, while subtracting a constant from the function (-2) shifts the graph downward.
Part B: Finding h(x)
To find h(x) when f(x) = 3x² - 2, we substitute x + 5 for x in f(x) and then subtract 2:
h(x) = 3(x + 5)² - 2 - 2 = 3(x + 5)² - 4. This represents the new function h(x) after the transformation.
Part C: Evaluate h(-3)
For the calculation of h(-3), we use the found expression of h(x) and substitute x with -3:
h(-3) = 3(-3 + 5)² - 4 = 3(2)² - 4 = 3(4) - 4 = 12 - 4 = 8. Hence, h(-3) evaluates to 8.